If your eigensystem is $\large \{ (\lambda_1, \vec{v_1}), (\lambda_2, \vec{v_2}) \}$ then the solution to the system of diff eqs is given by

$\large \begin{pmatrix}x_1(t) \\ x_2(t)\end{pmatrix} = c_1 \vec{v_1} e^{\lambda_1 t} + c_2 \vec{v_2} e^{\lambda_2 t}$

and you use the initial conditionals to solve for $c_1$ and $c_2$.